Increasing electricity demand makes it necessary to expand current electricity generation and transmission capacity. A multi-stage multi-scale linear stochastic programming model is proposed to seek an optimal investment plan in electricity generation technologies, storage and transmission lines over a long (multi-decade) time horizon. The multi-stage multi-scale framework allows decisions and uncertainties that occur at different points in time at different scales to be modeled. The stochastic framework allows the representation of uncertainties involving both investment and dispatch decisions. These features are very important given the long-term effect of each investment decision and possible uncertainties involved in this process.
With the goal of minimizing the total cost of satisfying demand, the model considers investment cost uncertainty in different power plants, fuel cost uncertainty, and uncertain long-run demand growth. It also provides a detailed treatment of the operating stage so that the effect of dispatch decisions on long-term investment decisions is captured. For example, some inter-temporal constraints such as ramping constraints and storage constraints are more properly modeled. Solving operating stages for each day over the long-term planning horizon in this model would be computationally expensive. Thus, we use representative days to represent daily operating conditions and only solve operating stage for those days. k-means clustering, hierarchical clustering, and dynamic time warping techniques are applied to find representative days. Three different methods based on these techniques are proposed and compared. The application of clustering methods helps simplify the amount of data in the operating stage while maintaining the modeling of representative and various daily demand, wind, and solar conditions.
The proposed model is large in size given its consideration of time horizon and uncertainties. The progressive hedging algorithm (PHA) is applied to decompose the stochastic model so that each scenario can be solved separately, greatly reducing the required computation time. The optimal solutions could be restored from solutions for each scenario. To assess the quality of the solutions generated from the PHA algorithm, we find that dual prices of the non-anticipativity constraints define implicit lower bounds for multistage stochastic problems, and the incumbent solutions given by the PHA provide upper bounds.
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